Effects of temperature dependent viscosity on peristaltic flow of a Jeffrey-six constant fluid in a non-uniform vertical tube

Hooman & Gurgenci (2008) Int. J. Heat Mass Trasnfer Volume 51, Issues 5-6, Pages 1139-1149

1

Effects of temperature dependent viscosity on Bénard convection in a porous

medium using a non-Darcy model

K. Hooman, H. Gurgenci

School of Engineering, The University of Queensland, Brisbane, Australia

Abstract

Temperature dependent viscosity variation effect on Bénard convection, of a gas or a

liquid, in an enclosure filled with a porous medium is studied numerically, based on the

general model of momentum transfer in a porous medium. The Arrhenius model, which

proposes an exponential form of viscosity-temperature relation, is applied to examine

three cases of viscosity-temperature relation: constant (µ=µC), decreasing (down to

0.13µC) and increasing (up to 7.39µC). Effects of fluid viscosity variation on isotherms,

streamlines, and the Nusselt number are studied. Application of the effective and average

Rayleigh number is examined. Defining a reference temperature, which does not change

with the Rayleigh number but increases with the Darcy number, is found to be a viable

option to account for temperature-dependent viscosity variation.

Keywords: Temperature-dependent viscosity, Natural convection, Porous medium,

Nusselt number, Bénard problem

Nomenclature

b viscosity variation number

CF inertia coefficient

Da the Darcy number, Da=K/L2

E error in calculating Nu based on effective/average Ra, /

/eff am

Nu Nu Nu−

eNu error in calculating Nu based on reference temperature approach

/Nu*Nu -Nu eNu =

maxψe error in calculating maxψ based on reference temperature approach

maxmaxmax /*- max

ψψψψ =e


2

g gravitational acceleration, m/s2

k porous medium thermal conductivity, W/m.K

K permeability, m2

L cavity height, m

Nu the Nusselt number

Nu* the Nusselt number with viscosity at reference temperature

P* pressure, Pa

Prc modified Prandtl number , αφν /Pr cc =

Ra Rayleigh-Darcy number, Ra=DaRaf

Raf the fluid Rayleigh number, ( )ανβ cCHf LTTgRa /)( 3−=

Sφ source term for ϕ equation

Sω source term for vorticity transport equation

T* temperature, K

u* x*-velocity, m/s

u u*L/ α

*U mean velocity 22 ** vu + , m/s

U dimensionless mean velocity 22 vu +

v* y*-velocity, m/s

v v*L/ α

x* horizontal coordinate, m

x x*/L

y* vertical coordinate, m

y y*/L

Greek symbols

α thermal diffiusivity of the porous medium, m2/s

β thermal expansion coefficient, 1/K

Γφ diffusion parameter, m2/s


3

Λ inertial parameter ( )KLC cF Pr/2φ=Λ

θ dimensionless temperature (T*-TC)/(TH-TC)

η kinematic viscosity ratio

µ fluid viscosity, N⋅s/m2

ρ fluid density, Kg/m3

υ kinematic viscosity, m2/s

φ generic variable

ψ stream-function

ψmax maximum value of stream-function

ψmax* ψmax with viscosity at reference temperature

φ porosity

ω vorticity

subscript

am arithmetic mean

ave average

C of cold wall

cp constant property

eff effective

H of hot wall

ref of reference temperature


4

1. Introduction

With interesting industrial applications such as filters and catalytic reactors,

underground contaminant transport, oil and gas exploration and extraction, and grain

storage, natural convection in porous media is a topic of increasing importance. The

buoyancy-induced flow in a cavity heated from below leads to patterns of convection

cells. The direction of fluid rotation alternates between neighboring cells. Known in the

literature as the Bénard convection, the fluid motion starts only when the imposed

temperature difference exceeds a certain value. The imposed temperature difference is

generally represented by the dimensionless Rayleigh number. The critical Rayleigh-

Darcy number, which indicates the onset of Bénard convection, is known to be equal to

4π2 for the Darcy flow in a porous medium bounded by two infinite horizontal isothermal

plates. This problem is sometimes referred to as the Darcy-Bénard problem.

Fundamentally, the momentum transport process in a porous medium is subject to

additional viscous and quadratic inertial effects, representing deviations from the familiar

Darcy law. The effects of the quadratic inertia and the viscous terms on natural

convection were investigated by Lauriat and Prasad [1], Kladias and Prasad [2], Khashan

et al. [3], and Lage [4]. On the other hand, the pioneering work of Vafai and Tien [5],

which was later revisited by Hsu and Cheng [6], is widely accepted for using the volume-

averaging technique coupled with semi-empirical formulas to arrive at the two-

dimensional momentum equation. Later reports of Merrikh and co-workers [7-9] have

elaborated on the application of the above method, to name a few.

Modeling heat transfer in a porous medium, in its turn, is a challenging problem.

Involving various presumptions and simplifications, formulating the thermal energy

equation is a continuous source of dispute and discussion as reflected in the large number

of papers on the topic [10-23].

Our review of literature has indicated that most of the reported studies on Bénard

convection assume constant viscosity. However, the fluid viscosity usually has a strong

dependence on temperature. For example, the viscosity of glycerin has a threefold

decrease in magnitude for a 10oC rise in temperature. This trend is not only observed in

highly viscous liquids, such as glycerin, but can also happen in other fluids such as water


5

where the viscosity decreases by about 240 percent when the temperature increases from

10oC to 50oC. Such severe changes in the fluid viscosity will result in different heat and

fluid flow patterns compared to constant property solutions [24]. Some authors (see for

example [25-28]) have investigated natural convection with temperature dependent

viscosity while keeping the other fluid properties constant (this assumption is known to

be valid for some fluids [29]).

A relatively important problem is the study of ore body formation and

mineralization in hydrothermal systems for which the temperature-dependent viscosity

variation should be considered as noted by Lin et al. [24] have reported analytical

solutions, backed by some numerical simulations, to claim that the viscosity variation

effects will destabilize the Darcy-Bénard convection. The reference viscosity adopted in

their Rayleigh-Darcy number was based on the cold wall conditions.

On the other hand, in a notable study, commenting on [25-27], Nield [30, 31]

argued that the effect of property variation on free convection is artificial and should

disappear if one uses an effective Rayleigh number based on mean values. Nield [31]

showed that, if the mean values are used, the critical Rayleigh number remains unaltered,

which indicates that the flow of a fluid with temperature-dependent viscosity is no less

stable than a constant-property one. The convection does not start at a smaller Rayleigh

number with a variable-property fluid as long as proper care is applied when calculating

the Rayleigh number. He also concluded that when the viscosity varied within one order

of magnitude, the concept of effective Rayleigh number would work while it was

conceded that possible localized flow in a part of the flow region might invalidate this

argument if the property variation were more severe. It is interesting to note that, in an

example of a fluid clear of solid material, for natural convection of corn syrup with a

temperature-dependent viscosity, even extreme viscosity variations, did not have a

significant effect on the overall heat transfer coefficient provided the properties were

evaluated at the mean temperature and a correction factor was used [32]. This conclusion

is in line with what was reported for natural convection of air in a square enclosure [33].

Siebers et al. [34] have come up with the same conclusion for laminar natural convection

of air along a vertical plate. Interestingly, they had to apply a correction factor on their

Nusselt number for more intense convection case with the flow becoming turbulent.


6

The problem becomes more complicated when one observes that Guo and Zhao

[28] evaluated the fluid properties at the arithmetic mean temperature (the mean of hot

and cold wall temperatures in a laterally heated box) but their results still showed

significant differences between constant- and variable-property flows. For example, for

Da=10-4 and Ra=10, the Nusselt number was about 75% higher than the constant

property case.

This gives us the impression that more work on the issue is called for. A

numerical simulation of the problem is presented here to investigate the effects of

temperature-dependent viscosity on natural convection in a square porous cavity. The

well-known problem of Bénard convection in a porous cavity is undertaken based on a

non-Darcy flow model similar to that of [9]. However, our work is different from the

previous studies addressing the variable viscosity effects on the Bénard convection as we

considered the general model including the viscous and (both quadratic and convective)

inertia terms. Several models have been used in the literature to account for the viscosity

variation with the temperature. Representing most common fluids, the Arrhenius model

proposes an exponential form of viscosity-temperature behavior and is reported to be

quite effective [35]. This model is applied here for flow of an incompressible gas or

liquid. The viscosity of a gas usually increases with temperature and the viscosity of a

liquid does the reverse. Both cases are considered here.

2. Model equations

Incompressible natural convection of a fluid with temperature-dependent viscosity

in a square enclosure filled with homogeneous, saturated, isotropic porous medium with

the Oberbeck–Boussinesq approximation for the density variation in the buoyancy term is

considered, as shown in Fig. 1. It is assumed that the solid matrix and the fluid are in

local thermal equilibrium. The equations that govern the conservation of mass,

momentum and energy can be written as follows

( * ) (v* )( ) ( )

* * * * * *

uS

x y x x y yϕ ϕ ϕϕ ϕ ϕ ϕ∂ ∂ ∂ ∂ ∂ ∂+ = Γ + Γ +

∂ ∂ ∂ ∂ ∂ ∂ (1)


7

where ϕ stands for the dependent variables u*, v*, T*; and ϕΓ , ϕS are the corresponding

diffusion and source terms, respectively, for the general variable ϕ , as summarized in

Table 1. Other parameters are defined in the nomenclature.

The following exponential variation in kinematic viscosity ratio (with temperature) is

assumed

( )expc

bνη θν

= = , (2)

where the viscosity variation number, b, is positive/negative in case of a gas/liquid whose

viscosity increases/decreases with an increase in temperature. The cold wall condition is

assumed as our reference state so that νc is the kinematic viscosity measured at Tc. Our

dimensionless temperature is θ=(T*-TC)/(TH-TC). One also notes that the Taylor series

expansion for very small values of b leads to linear or inverse linear relations for

viscosity with temperature as

( )( ),1

11

,1

θνν

θνν

b

b

c

c

−=

+= (3-a,b)

similar to the models applied in [36-39].

The dimensionless stream-function is defined as

,

v .

uy

x

ψ

ψ

∂=∂

∂= −∂

(4-a,b)

With this definition, the continuity equation is satisfied identically. The dimensionless

coordinates are (x,y)=(x*,y*)/L and the velocity components are (u,v)=(u*, v*)(L/ α).

Taking the curl of x*- and y*-momentum equations and eliminating the pressure terms,

one finds the dimensionless vorticity transport equation as

( )( )2. Pr / bc wu Da e U Sθω ω ω ω∇ = ∇ − − Λ + (5)

where


8

2 2 2 2

2 2

/

.

w f

U US Da Ra

x x y y x x y y x

y x x y y y x x x y x y

η ψ η ψ ψ ψ θ

η ψ η ψ η ψ η ψ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + Λ + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂− + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(6)

The Rayleigh-Darcy number, or simply Ra hereafter, is defined as Ra=DaRaf.

The vorticity directed in z direction is defined as

∂∂+

∂∂−=

2

2

2

2

yx

ψψω . (7)

The thermal energy equation now takes the following form

2.u θ θ∇ = ∇ . (8)

The average Nusselt number as the ratio of the actual heat transfer to that of pure

conduction is defined as [3]

∫ ∂∂=

1

0.

)0,(dx

y

xNu

θ (9-a)

The problem is now to solve Eqs. (5-9) subject to no-slip boundary condition on the

walls, i.e. u=v=0, and the following thermal boundary conditions

0; vertical walls,

0; top wall,

1; bottom wall.

x

θ

θθ

∂ =∂

==

(9-b-d)

3. Numerical details

Numerical solutions to the governing equations for vorticity, stream-function, and

dimensionless temperature are obtained by finite difference method, using the Gauss-

Seidel technique with SOR. The governing equations are discretized by applying second-

order accurate central difference schemes. For the numerical integration, algorithms

based on the trapezoidal rule are employed similar to [40]. Details of the vorticity-stream-

function method, and applied boundary conditions may be found in [41] and are not

repeated here.

All runs were performed on a 61 x 61 grid. The Darcy number ranges from 10-6 to

10-3 while the reference Prandtl number is fixed at unity similar to Merrikh and Mohamad


9

[9]. The inertia coefficient, CF is fixed at 0.56 similar to Lage [4]. Grid independence was

verified by running different combinations of Da, Raf, and b on three different grid sets

41x41, 61x61 and 91x91. Less than 1% difference between results obtained on different

grids is observed. The convergence criterion (maximum relative error in the values of the

dependent variables between two successive iterations) in all runs was set at 10-5.

A test on the accuracy of the numerical procedure is provided by comparing the

results against those for special cases quoted in the literature, i.e. [42-45]. This

comparison for the average Nusselt number and the maximum stream-function value is

shown in Tables 2 and 3, respectively.

4. Results and Discussion

Figures 2 and 3 are designed to reflect the effects of the key parameters (being b,

Da, Ra, and Raf) on isotherms and streamlines. The porous-medium Rayleigh number,

Ra, is 50 and 300, respectively, for Figures 2 and 3. Both extreme positive and negative

values of b are included to represent fluids with viscosities increasing and decreasing

with temperature. The results of isotherms and streamlines for different values of Da

(Da=10-3 and 10-4) are plotted on different charts in each figure. To maintain a constant

Ra value, the value of Raf is altered along with Da. One can easily see that with negative

values of b, representing viscosity decreasing with an increase in temperature, the flow

patterns are stronger. On the other hand, the converse can be deduced with positive

values of b. The constant property solution is found to be somewhere between the two

cases, as expected. In all of our contour plots the contours are plotted at equal increments

of the plotted variable. Comparing Figs. 2 and 3, it is clear that with a fixed value of Da,

an increase in either Ra or Raf leads to stronger convective flows, as expected. Examining

the streamlines, which are normalized by maxψ , it is quite clear that with positive values

of b the core region moves toward the cold wall while with positive counterparts this

region tends to be stretched downward to form an elliptical pattern and this elliptical

pattern is more identifiable for Ra=300. Moreover, with this Rayleigh number, moving

from constant property to b=-2, the change in the size of the core region is less than the

one associated with the change in the opposite direction, i.e. from b=0 to b=2. For Ra=50


10

and b=2, with either values of Da=10-3 or 10-4, the isotherms are nearly horizontal

implying that there is no convection flow. On the other hand, with b =-2 compared to the

other two values of b, regardless of Ra and Da values, the convection patterns are

stronger and isotherms are more stretched towards the horizontal walls.

Fig. 4 shows the line diagrams of the dimensionless horizontal mid-plane

velocity, v(x,0.5), when b varies from -2 to 2 with Da=10-3 and for two cases of Ra=50

and 300. As expected, a higher value of Ra promotes mixing and this is manifested as an

increase in the maximum vertical velocity. It is interesting to note that with b=2 the flow

nearly subsides while for b=-2 the peak is nearly five times higher than that of the

constant property case. However, for Ra=300, the ratio of the velocity peaks is not that

high and it figures out at 1.5, approximately.

Fig. 5 shows the dependence of Nu and maxψ on b for different values of Da and

Ra. A Nu value of 1 means the actual heat transfer being due to conduction only, i.e. Nu

only exceeds 1 when there is convection. As seen, both Nu and maxψ decrease with an

increase in the absolute value of b. It is interesting that with Ra=50, for which a

convective flow pattern is expected based on constant property solutions, with positive b

values of 0.1, 0.4, and 0.5 the flow nearly subsides, for Da values of 10-3, 10-4, and 10-6,

respectively. However, for Ra=100 the value of b needs to be as high as 1.7 for the same

phenomenon to occur. It is observed that increasing Ra, raises the Nu level but,

interestingly, moving to other Ra values with a fixed Da, the slope of Nu-b plots will

remain almost the same. Interestingly, maxψ shows similar behavior; however, it is

observed that for the lowest Darcy value, Da=10-6, the b−maxψ curve becomes a concave

one instead of the convex distribution formed for higher Da values.

Based on the observation that the Nu-b plots are parallel for a fixed Da with

changing Ra, it is tempting to argue that defining an average Rayleigh number, the Nu-Ra

relation could remain, to a good approximation, independent of the changes in viscosity.

In the preceding discussion, the Rayleigh numbers were calculated at the cold wall

temperature. The apparent destabilizing effect of decreasing viscosity was observed in all

figures when the Rayleigh number was calculated this way. Let us now see what happens

when an average/effective Rayleigh number is used. It is instructive to note that there are


11

two approaches to account for variable property (forced or natural convection) problems.

The first one is evaluating the fluid property at the film temperature (arithmetic mean

value of maximum and minimum temperatures). The second one is evaluating the fluid

property at a reference temperature and using a correction factor to account for property

variations. More details may be found in Kakaç and Yener [29].

Nield [30] recommends using a harmonic average for the fluid viscosity in the

effective Rayleigh number. Since the Rayleigh number is inversely proportional to

viscosity, we define our effective Rayleigh number as the arithmetic mean of the

Rayleigh numbers at two extreme temperatures

+=2

HCeff

RaRaRa . (10)

The subscripts ‘H’ and ‘C’ are applied to show that heated and cooled wall temperatures

are applied to evaluate the viscosity. One notes that RaC=Ra, as applied so far, and that

using Eq. (2) one has

( )

−+=2

exp1 bRaRaeff . (11)

The effective Rayleigh numbers calculated by the above equation are shown in our Table

4 as Case 1.

On the other hand, Guo and Zhao [28] proposed the arithmetic mean temperature

as the reference temperature and evaluated the viscosity at that temperature. However,

when using this mean temperature, the Nusselt number showed notable differences from

the constant property case. This behavior could be expected, to some extent, in the light

of [32], where the authors recommended, for the clear fluid case, adding a viscosity

fraction to the constant property Nu-Ra correlations to make them useful in variable

property cases.

All in all, for this case, the average Rayleigh number reads

( )exp 0.5amRa Ra b= − (12)

wherein Raam is the Rayleigh number with the viscosity being evaluated at the arithmetic

mean temperature and is referred to as Case 2 in Table 4.

Using the Taylor series, it is an easy task to show that for small b values both of the two

approaches lead to the same answer being


12

( )1 0.5am effRa Ra Ra b= = − (13)

Nonetheless, for higher values of b the two methods will lead to very different results as

shown in Table 4 which lists the ratio of the variable property Nusselt number divided by

that of constant property, Nu/Nucp, versus average/effective Rayleigh number. As seen,

the results are closer for small values of b, however, increasing b not only the two

methods will diverge but also they lead to erroneous results compared to our numerical

solutions. It could be concluded that the concept of an effective Rayleigh number, though

proven to be useful to show the onset of convection for a porous layer heated form below,

is restricted to the case where an inverse linear viscosity-temperature relation is assumed

(and is equivalent to our model with very small b according to Eq. (3)). On the other

hand, the average Rayleigh number approach leads to better results for low Ra and b

cases and increasing either of the two parameters restricts the application of this method.

According to Table 4, none of the above methods are accurate and there is a need for

another alternative.

The issue is finding a reference temperature to evaluate the viscosity so that the

results will be valid for the entire b-domain that is considered in this analysis. Based on

our numerical results, it is reasonable to expect this reference temperature to change with

the porous medium permeability, which may be represented by the Darcy number. By

observation of the results, we have found this reference temperature to change with the

Darcy number as follows

6

4

3

0.45( ) 10 ,

0.4 ( ) 10 ,

0.35( ) 10 .

ref C H C

ref C H C

ref C H C

T T T T for Da

T T T T for Da

T T T T for Da

−

−

−

= + − =

= + − =

= + − =

(14-a,b,c)

Substitution of the above reference temperature in Eq. (2), will lead to the following

average Rayleigh numbers

6

4

3

exp( 0.45 ) 10 ,

exp( 0.4 ) 10 ,

exp( 0.35 ) 10 .

ave C

ave C

ave C

Ra Ra b for Da

Ra Ra b for Da

Ra Ra b for Da

−

−

−

= − =

= − =

= − =

(15-a,b,c)

Table 5 is designed to show the results of our constant property calculation with viscosity

being evaluated at the above reference temperature. It seems that our predictions are


13

within good agreement with the maximum error of 10% for Nu and 12% for ψmax for the

extreme viscosity variation cases. It may be concluded that one can still apply the

constant property solutions available in the literature with the only modification that the

fluid property is evaluated at the reference temperature recommended here. Another point

worthy of comment is that our results are limited within a range of the Darcy numbers

being those relevant to clear fluid (1/Da→0) and Darcy flow model (Da→0). For these

two cases the reference temperatures are Tref= TC+0.5(TH-TC) and Tref= TC+0.25(TH-TC)

with the former being recommended indirectly by Nield [30] (for small values of b) for

the Darcy flow model and the latter proposed by Zhong et al. [33] for the clear fluid

natural convection in a laterally heated box. It is interesting that though the flow structure

is completely different in a lateral and bottom heating case, as noted by Nield [46] and

implied by Bejan [41], the limiting reference temperature for the clear fluid case is the

same. The dependence of the reference temperature on the Darcy number is expected as

each Da value is associated with a unique convection pattern. For the sake of simplicity,

we propose a rough and ready estimation for the dependence of the reference temperature

on the Darcy number as follows

( )0.150.5 1 0.848 ( ) ref C H CT T Da T T= + − − (16)

The average Rayleigh number, Eq. (15), now takes the following form

( )( )0.15exp 0.5 1 0.848ave CRa Ra b Da= − − (17)

However, one should be warned that these last two equations are valid for the range of

the Darcy number considered in our study being 10-3-10-6. One notes that for small values

of b with Da=0 the average Rayleigh number tends to the effective Rayleigh number of

Nield [30].

5. Conclusion

Numerical simulation of Bénard natural convection in a bottom heated porous-

saturated square enclosure is presented based on the general momentum equation. The

Arrhenius model for the variation of viscosity with the temperature is applied. A

reference temperature approach is undertaken to account for viscosity variation. It is


14

found that the reference temperature, at which the fluid properties should be evaluated, is

an increasing function of the Darcy number and is approximately independent of the

other parameters considered here. Applying this reference temperature, one can still use

the constant property results and this, in turn, will reduce the computational time and

expense required for solving a variable property problem.

Acknowledgments

The first author, the scholarship holder, acknowledges the support provided by The

University of Queensland in terms of UQILAS, Endeavor IPRS, and School Scholarship.

References

[1] Lauriat G, Prasad V. Non-Darcian Effects on Natural-Convection in a Vertical Porous Enclosure. International Journal of Heat and Mass Transfer 1989;32:2135. [2] Kladias N, Prasad V. Flow Transitions in Buoyancy-Induced Non-Darcy Convection in a Porous-Medium Heated from Below. Journal of Heat Transfer-Transactions of the Asme 1990;112:675. [3] Khashan SA, Al-Amiri AM, Pop I. Numerical simulation of natural convection heat transfer in a porous cavity heated from below using a non-Darcian and thermal non-equilibrium model. International Journal of Heat and Mass Transfer 2006;49:1039. [4] Lage JL. Effect of the Convective Inertia Term on Benard Convection in a Porous-Medium. Numer. Heat Tranf. A-Appl. 1992;22:469. [5] Vafai K, Tien CL. Boundary and Inertia Effects on Flow and Heat-Transfer in Porous-Media. International Journal of Heat and Mass Transfer 1981;24:195. [6] Hsu CT, Cheng P. Thermal Dispersion in a Porous-Medium. International Journal of Heat and Mass Transfer 1990;33:1587. [7] Merrikh AA, Lage JL, Mohamad AA. Natural convection in nonhomogeneous heat-generating media: Comparison of continuum and porous-continuum models. J. Porous Media 2005;8:149. [8] Merrikh AA, Mohamad AA. Transient natural convection in differentially heated porous enclosures. J. Porous Media 2000;3:165. [9] Merrikh AA, Mohamad AA. Non-Darcy effects in buoyancy driven flows in an enclosure filled with vertically layered porous media. International Journal of Heat and Mass Transfer 2002;45:4305. [10] Beckermann C, Viskanta R, Ramadhyani S. A Numerical Study of Non-Darcian Natural-Convection in a Vertical Enclosure Filled with a Porous-Medium. Numerical Heat Transfer 1986;10:557. [11] Figueiredo JR, Llagostera J. Comparative study of the unified finite approach exponential-type scheme (UNIFAES) and its application to natural convection in a porous cavity. Numer Heat Tranf. B-Fundam. 1999;35:347. [12] Guo ZL, Zhao TS. A lattice Boltzmann model for convection heat transfer in porous media. Numer Heat Tranf. B-Fundam. 2005;47:157.


15

[13] Hirata SC, Goyeau B, Gobin D, Cotta RM. Stability of natural convection in superposed fluid and porous layers using integral transforms. Numer Heat Tranf. B-Fundam. 2006;50:409. [14] Kim GB, Hyun JM. Buoyant convection of a power-law fluid in an enclosure filled with heat-generating porous media. Numer. Heat Tranf. A-Appl. 2004;45:569. [15] Kumar BVR, Shalini. Natural convection in a thermally stratified wavy vertical porous enclosure. Numer. Heat Tranf. A-Appl. 2003;43:753. [16] Mansour A, Amahmid A, Hasnaoui M, Bourich M. Multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and submitted to horizontal concentration gradient in the presence of soret effect. Numer. Heat Tranf. A-Appl. 2006;49:69. [17] Mojtabi MCC, Razi YP, Maliwan K, Mojtabi A. Influence of vibration on soret-driven convection in porous media. Numer. Heat Tranf. A-Appl. 2004;46:981. [18] Prasad V, Tuntomo A. Inertia Effects on Natural-Convection in a Vertical Porous Cavity. Numerical Heat Transfer 1987;11:295. [19] Slimi K, Mhimid A, Ben Salah M, Ben Nasrallah S, Mohamad AA, Storesletten L. Anisotropy effects on heat and fluid flow by unsteady natural convection and radiation in saturated porous media. Numer. Heat Tranf. A-Appl. 2005;48:763. [20] Slimi K, Zili-Ghedira L, Ben Nasrallah S, Mohamad AA. A transient study of coupled natural convection and radiation in a porous vertical channel using the finite-volume method. Numer. Heat Tranf. A-Appl. 2004;45:451. [21] Vasseur P, Wang CH, Sen M. The Brinkman Model for Natural-Convection in a Shallow Porous Cavity with Uniform Heat-Flux. Numerical Heat Transfer 1989;15:221. [22] Beji H, Gobin D. Influence of Thermal Dispersion on Natural-Convection Heat-Transfer in Porous-Media. Numer. Heat Tranf. A-Appl. 1992;22:487. [23] Al-Amiri AM. Natural convection in porous enclosures: The application of the two-energy equation model. Numer. Heat Tranf. A-Appl. 2002;41:817. [24] Lin G, Zhao CB, Hobbs BE, Ord A, Muhlhaus HB. Theoretical and numerical analyses of convective instability in porous media with temperature-dependent viscosity. Commun. Numer. Methods Eng. 2003;19:787. [25] Jang JY, Leu JS. Buoyancy-Induced Boundary-Layer Flow of Liquids in a Porous-Medium with Temperature-Dependent Viscosity. Int. Commun. Heat Mass Transf. 1992;19:435. [26] Jang JY, Leu JS. Variable Viscosity Effects on the Vortex Instability of Free-Convection Boundary-Layer Flow over a Horizontal Surface in a Porous-Medium. International Journal of Heat and Mass Transfer 1993;36:1287. [27] Kassoy DR, Zebib A. Variable Viscosity Effects on Onset of Convection in Porous-Media. Phys. Fluids 1975;18:1649. [28] Guo ZL, Zhao TS. Lattice Boltzmann simulation of natural convection with temperature-dependent viscosity in a porous cavity. Prog. Comput. Fluid Dyn. 2005;5:110. [29] Kakaç S, Yener Y. Convective heat transfer Boca Raton: CRC Press, 1995. [30] Nield DA. Estimation of an Effective Rayleigh Number for Convection in a Vertically Inhomogeneous Porous-Medium or Clear Fluid. International Journal of Heat and Fluid Flow 1994;15:337.


16

[31] Nield DA. The effect of temperature-dependent viscosity on the onset of convection in a saturated porous medium. Journal of Heat Transfer-Transactions of the Asme 1996;118:803. [32] Chu TY, Hickox CE. Thermal-Convection with Large Viscosity Variation in an Enclosure with Localized Heating. Journal of Heat Transfer-Transactions of the Asme 1990;112:388. [33] Zhong ZY, Yang KT, Lloyd JR. Variable Property Effects in Laminar Natural-Convection in a Square Enclosure. Journal of Heat Transfer-Transactions of the Asme 1985;107:133. [34] Siebers DL, Moffatt RF, Schwind RG. Experimental, Variable Properties Natural-Convection from a Large, Vertical, Flat Surface. Journal of Heat Transfer-Transactions of the Asme 1985;107:124. [35] Harms TM, Jog MA, Manglik RM. Effects of temperature-dependent viscosity variations and boundary conditions on fully developed laminar forced convection in a semicircular duct. Journal of Heat Transfer-Transactions of the Asme 1998;120:600. [36] Nield DA, Kuznetsov AV. Effects of temperature-dependent viscosity in forced convection in a porous medium: Layered-medium analysis. J. Porous Media 2003;6:213. [37] Nield DA, Porneala DC, Lage JL. A theoretical study, with experimental verification, of the temperature-dependent viscosity effect on the forced convection through a porous medium channel. Journal of Heat Transfer-Transactions of the Asme 1999;121:500. [38] Hooman K. Entropy-energy analysis of forced convection in a porous-saturated circular tube considering temperature-dependent viscosity effects. International Journal of Exergy 2006;3:436–451. [39] Hooman K, Gurgenci H. Effects of temperature-dependent viscosity variation on entropy generation, heat, and fluid flow through a porous-saturated duct of rectangular cross-section. Applied Mathematics and Mechanics (English edition) 2007;28:69 [40] Hooman K. A perturbation solution for forced convection in a porous saturated duct. Journal of computational and applied mathematics 2007;in press (doi: 10.1016/j.cam.2006.11.005). [41] Bejan A. Convection heat transfer. Hoboken, N.J. : Wiley, 1984. [42] Prasad V, Kulacki FA. Natural-Convection in Horizontal Porous Layers with Localized Heating from Below. Journal of Heat Transfer-Transactions of the Asme 1987;109:795. [43] Caltagirone JP. Thermoconvective Instabilities in a Horizontal Porous Layer. J. Fluid Mech. 1975;72:269. [44] Schubert G, Straus JM. 3-Dimensional and Multicellular Steady and Unsteady Convection in Fluid-Saturated Porous-Media at High Rayleigh Numbers. J. Fluid Mech. 1979;94:25. [45] Bilgen E, Mbaye M. Benard cells in fluid-saturated porous enclosures with lateral cooling. International Journal of Heat and Fluid Flow 2001;22:561. [46] Nield DA. The Modeling of Viscous Dissipation in a Saturated Porous Medium. Journal of Heat Transfer 2006;in press.


17

Table 1 Summary of the solved governing equations Equations ϕ

ϕΓ ϕS

Continuity 1 0 0

x*-momentum u*/φ ν 1/ 2

* *1 * *

*FC u Up u

x K K

φνρ

∂− − −∂

y*-momentum v*/ φ ν ( )1/2

* *1 * **

*F

c

C v Up vg T T

y K K

φν βρ

∂− − − + −∂

Energy T* α 0

Table 2 Present Nu values for Da=10-6 versus those in the literature for the Darcy model.

Ra Present Ref. [45] Ref. [42] Ref. [43] Ref. [4] (Da=10-6)

50 1.464 1.443 1.45 - 1.44

100 2.643 2.631 2.676 2.651 2.62

200 3.782 3.784 3.813 3.808 3.762

250 4.15 4.167 - - 4.139

300 4.456 4.487 - 4.514 -

Table 3 Present ψmax values for Da=10-6 versus those in the literature for the Darcy

model.

Ra Present Ref. [45] Ref. [43]

50 2.096 2.092 2.112

100 5.319 5.359 5.377

200 8.845 8.931 8.942

250 10.131 10.244 10.253

300 11.252 11.394 11.405


18

Table 4-A Calculation of the effective and average Rayleigh numbers and Nu/Nucp for

(Da=10-3, Ra=50)

b Numerical Case 1 Case 2

Nu / Nucp Raeff Nu

/Nucp

E % Raam Nu /

Nucp

E %

-2 1.845 209.7

3

2.517 36.42 135.9

2

2.07 12.1

7

-0.5 1.266 66.22 1.327 4.82 64.2 1.291 1.97

0.5 0.899 40.16 0.9 0.11 38.94 0.893 0.67

Table 4-B Calculation of the effective and average Rayleigh numbers and Nu/Nucp for

(Da=10-4, Ra=100)

b Numerical Case 1 Case 2

Nu/Nucp Raeff Nu/Nucp E % Raam Nu/Nucp E %

-2 1.4264 419.45 1.8219 27.73 271.83 1.579 10.72

-1 1.25 185.91 1.362 8.9 164.87 1.2922 3.35

1 0.703 68.39 0.766 8.9 60.65 0.688 2.1

Table 5 Application of the reference temperature approach adopted here for some values

of Da, Ra, and b.

Da Ra b Raave Nu* Nu eNu% maxψ * maxψ

maxψe

%

10-6

50 -2 122.98 2.963 3.013 1.66 6.486 7.206 9.99

-1 78.42 2.223 2.308 3.68 4.162 4.505 7.61

100

-2 245.96 4.096 4.01 2.14 9.991 11.162 10.49

-1 156.83 3.359 3.395 1.06 7.483 7.882 5.33

1 63.76 1.863 1.813 2.71 3.2 3.128 5.06

200

-2 491.92 5.248 5.02 4.54 14.51 15.797 8.15

-1 313.66 4.498 4.486 0.27 11.429 11.911 4.05

1 127.53 3.02 2.93 2.93 6.445 6.38 1.01


19

2 81.31 2.283 2.16 5.71 4.329 4.423 2.12

300

-1 470.49 5.177 5.14 0.71 14.173 14.702 3.59

1 191.29 3.684 3.58 2.91 8.533 8.486 0.55

2 121.97 2.95 2.77 6.5 6.227 6.416 2.94

10-4

50 -2 111.28 2.584 2.62 1.37 5.32 5.928 10.23

-1 74.59 1.994 2.08 4.12 3.624 3.984 9.04

100

-2 222.55 3.563 3.47 2.86 8.536 8.941 4.53

-1 149.18 3.004 3.04 1.2 6.629 6.933 4.38

1 67.03 1.828 1.71 6.89 3.156 2.878 9.66

200

-2 445.11 4.507 4.34 3.85 12.287 12.4 0.91

-1 298.36 3.974 3.953 0.54 10.07 10.272 1.97

1 134.06 2.84 2.74 3.64 6.169 5.941 3.83

2 89.87 2.274 2.1 8.29 4.404 4.124 6.79

300

-1 447.55 4.514 4.471 0.96 12.318 12.465 1.18

1 201.1 3.414 3.31 3.14 8.009 7.863 1.85

2 134.8 2.84 2.61 8.8 6.169 5.981 3.15

10-3

50 -2 100.69 2.027 2.086 2.83 3.981 4.374 8.98

-1 70.95 1.592 1.69 5.8 2.74 3.011 9

100

-2 201.38 2.79 2.758 1.16 6.622 6.769 2.17

-1 141.91 2.417 2.459 1.71 5.276 5.495 3.99

1 70.47 1.587 1.442 10 2.612 2.375 9.97

2 49.66 1.11 1.03 7.77 0.911 0.824 10

200

-2 402.76 3.505 3.412 2.72 9.619 9.44 1.9

-1 283.81 3.151 3.156 0.2 8.074 8.168 1.15

1 140.94 2.41 2.305 4.56 5.27 4.986 5.69

2 99.32 2.01 1.827 10 3.901 3.535 10.35

300

-1 425.72 3.559 3.551 0.25 9.873 9.885 0.12

1 211.41 2.846 2.755 3.3 6.486 6.633 2.22

2 148.98 2.469 2.244 10 5.463 5.037 8.46


20

FIGURE CAPTIONS:

Fig. 1 Schematic of the problem considered

Fig. 2-a-d Isotherms and streamlines for Ra=50 with Da=10-3 and 10-4

Fig. 3-a-d Isotherms and streamlines for Ra=300 with Da=10-3 and 10-4

Fig. 4 The dimensionless horizontal mid-plane velocity versus x with some values of b for

Da=10-3 a)Ra=50, b)Ra=300

Fig. 5-a,b Plots of Nu and maxψ versus b for different values of Da and Ra.

y*, v*

x*,u*

Porous medium

* * 0, * Hu v T T= = =

* * 0

*0

*

u v

T

x

= =∂ =∂

* * 0

*0

*

u v

T

x

= =∂ =∂

g

L

L

* * 0, * Cu v T T= = =

Fig. 1 Schematic of the problem under consideration


21

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.9

0.1

Fig. 2-a) streamlines for Ra=50 and Da=0.001, Raf=50,000 (for figures2-3 dashed, solid, and dash-dotted

lines represent b=-2, 0, and 2, respectively)

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2-b) Isotherms for Ra=50 and Da=0.001, Raf=50,000


22

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.9

0.1

Fig. 2-c) streamlines for Ra=50 and Da=0.0001, Raf=500,000

Fig. 2-d) Isotherms for Ra=50 and Da=0.0001, Raf=500,000


23

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.9

0.1

Fig. 3-a) Streamlines for Ra=300 and Da=0.001, Raf=300,000

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.9

0.1

Fig. 3-b) Isotherms for Ra=300 and Da=0.001, Raf=300,000


24

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.9

0.1

Fig. 3-c) Streamlines for Ra=300 and Da=0.0001, Raf=3,000,000

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.9

Fig. 3-d) Isotherms for Ra=300 and Da=0.0001, Raf=3,000,000


25

x

v

0 0.25 0.5 0.75 1

-10

-5

0

5

10

15

b=-2b=0b=2

Fig. 4-a The dimensionless horizontal mid-plane velocity versus x for Ra=50 and Da=10-3.

x

v

0 0.25 0.5 0.75 1

-40

-30

-20

-10

0

10

20

30

40

50

b=-2b=0b=2

Fig. 4-b The dimensionless horizontal mid-plane velocity versus x for Ra=300 and Da=10-3.


26

b

Nu

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

1.5

2

2.5

3

3.5

4

4.5

5

5.51/Da=103

1/Da=104

1/Da=106

Ra=50

Ra=100

Ra=200

Ra=300

a)

b

Max

imu

mS

trea

mfu

nctio

n

-2 -1 0 1 2

2

4

6

8

10

12

14

16

18

1/Da=103

1/Da=104

1/Da=106

Ra=50

Ra=300

Ra=200

Ra=100

b)

Fig. 5-a,b Plots of Nu and maxψ versus b for different values of Da and Ra.

Effects of temperature dependent viscosity on peristaltic flow of a Jeffrey-six constant fluid in a non-uniform vertical tube

Documents